Unit I: Number Systems

Unit Diagnostic Test

1. Place value of 2 in 47,205 is 2 × 100 = 200

2. Word form: Thirty-eight thousand nine hundred

Expanded notation: 38,900 = 30,000 + 8000 + 900

3. XC + LX = 90 + 60 = 150 = CL
4.
a. b.

79396 ÷ 23 = 3452

12686 ÷ 51 = Q: 248, R: 38
b.

19687 ÷ 35 = Q: 562, R: 17
5. Selling price of 12 books = ₹680
Profit on selling 12 books = ₹104
Cost price = Selling price – Profit
Cost price of 12 books = ₹680 – ₹104 = ₹576
Cost price of one book = ₹576 ÷ 12 = ₹48
Thus, cost price of one book is ₹48.

6. 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, 4 × 4 = 16, 4 × 5 = 20
So, the first five multiples of 4 are 4, 8, 12, 16, and 20.

7.

10 = 2 × 5 12 = 2 × 2 × 3
HCF of 10 and 12 = 2
LCM of 10 and 12 = 2 × 2 × 3 × 5 = 60
8.
3
a. Total number of parts = 4, Number of shaded parts = 3, Fraction =
4
1
b. Total number of parts = 7, Number of shaded parts = 1, Fraction =
7
2
c. Total number of parts = 4, Number of shaded parts = 2, Fraction =
4

2×2 4 2×3 6
9. = ; =
3×2 6 3×3 9
2 4 6
Two equivalent fractions of are and .
3 6 9
Note: More equivalent fractions are possible.

2 3
10. a.
3 4
LCM of 3 and 4 = 12
2 2×4 8 3 3×3 9
= = and = 4 × 3 = 12
3 3×4 12 4
8 9
<
12 12
2 3
So, < .
3 4

5 10
b.
6 12
LCM of 6 and 12 is 12.
5 5×2 10
= =
6 6×2 12
10 10
=
12 12
5 10
So, = .
6 12
 8 8
c.
9 11
LCM of 9 and 11 is 99.
8 8×11 88 8 8×9 72
= = and = =
9 9×11 99 11 11×9 99
88 72
>
99 99
8 8
So, > .
9 11

11. Like decimals are those decimals that have same number of decimal places (after decimal point).
So, 12.123, 12.135, and 16.235 are like fractions.

12.

0.325 ÷ 5 = 0.065
2 17 3 17−9 8 2
13. a. 5 − 3 = − = = =2
3 3 1 3 3 3
1 36 3 36−15 21
b. 7 − 3 = − = =
5 5 1 5 5
3 7 66 7 (66×9)−(7×7) 594−49 545
c. 9 − = − = = =
7 9 7 9 63 63 63

14.

Difference between 4168 and 1932 is 2236.

15. To find the required number we must subtract 3571 from 9000.
The required number = 9000 – 3571 = 5429
Thus, 5429 must be subtracted from 9000 to get 3571 as the difference.
 1 Numbers

Try This!(Page no. 9)

a. 9999999 + 1 = 10,000,000
b. 5 zeros
c. 6 zeros
d. 1 million = 1,000,000 = 1 followed by 6 zeros = 10 lakhs

Exercise 1A
1.

a. 6,00,00,000 + 50,00,000 + 7,00,000 + 40,000 + 3000 + 600 + 80 + 3 = 65,743,683

b. 5,00,000 + 40,000 + 7000 + 800 + 60 + 4 = 5,47,864
c. 30,00,000 + 2,00,000 + 40,000 + 8000 + 700 + 2 = 32,48,702
d. 40,00,000 + 60,000 + 3000 + 60 + 8 = 40,63,068
e. 7,00,00,000 + 40,000 + 400 + 9 = 7,00,40,409
f. 3,00,00,000 + 7,00,000 + 800 + 5 = 3,07,00,805
g. 40,00,000 + 7,00,000 + 60 + 3 = 47,00,063
h. 8,00,00,000 + 4,00,000 + 70,000 + 600 + 40 + 7 = 8,04,70,647
2.

a. 67,43,859 = 60,00,000 + 7,00,000 + 40,000 + 3000 + 800 + 50+9

b. 4,32,087 = 4,00,000 + 30,000 + 2000 + 80 + 7
c. 23,80,043 = 20,00,000 + 3,00,000 + 80,000 + 40 + 3
d. 3,12,45,678 = 3,00,00,000 + 10,00,000 + 2,00,000 + 40,000 + 5000 + 600 + 70 + 8
e. 6,54,837 = 6,00,000 + 50,000 + 4000 + 800 + 30 + 7
f. 78,50,005 = 70,00,000 + 8,00,000 + 50,000 + 5
g. 75,06,080 = 70,00,000 + 5,00,000 + 6000 + 80
h. 7,32,80,054 = 7,00,00,000 + 30,00,000 + 2,00,000 + 80,000 + 50 + 4
3.
a. The value of 8 in 4,57,832 is 8 × 100 = 800.
b. The value of 6 in 85,93,061 is 6 × 10 = 60.
c. The value of 4 in 4,32,75,945 is 4 × 1,00,00,000 = 4,00,00,000.
d. The value of 4 in 54,985 is 4 × 1000 = 4000.
e. The value of 9 in 1,92,837 is 9 × 10,000 = 90,000.
f. The value of 8 in 48,79,614 is 8 × 1,00,000 = 8,00,000.
g. The value of 5 in 3,54,78,463 is 5 × 10,00,000 = 50,00,000.
h. The value of 8 in 6,57,298 is 8 × 1 = 8.
4.

a. Five lakh forty thousand six hundred fifty: 5,40,650

b. Fifteen lakh thirty-six thousand four hundred thirty-nine: 15,36,439
c. Five crore fifty-four lakh twenty-seven thousand eight hundred sixty-nine: 5,54,27,869
d. Six crores five lakh forty-three thousand sixty-five: 6,05,43,065
e. Eighty-three lakh forty-four thousand seven: 83,44,007
f. Seventy-eight lakh six thousand five hundred six: 78,06,506
g. Eight crore seven lakh eighty-six thousand sixty-eight: 8,07,86,068
h. Four crore seven hundred: 4,00,00,700
5.
a. Six lakh eighty-seven thousand three hundred fifty-four
b. Six lakh eighty thousand six hundred thirty-six
c. Forty lakh forty-nine thousand nine hundred seventy-four
d. Forty-eight lakh thirty-two thousand seven hundred forty-two
e. Thirty lakh seventy thousand seven
f. Five crore thirty-three lakh seventy-four thousand six
g. Six crore fifty-four lakh eighty-three thousand nine
h. Five crore four lakh ninety thousand seven hundred five
6.
a. 5,42,867
b. 97,53,110
c. 36,91,215
d. 2,46,81,012
7. Place value of 9 in 4,59,326 is 9 × 1000 = 9000
Face value of 9 in 4,59,326 = 9
Difference between the place value and face value = 9000 – 9 = 8991

Try This! (Page 12)

1. 8,98,684 < 89,97,585 < 89,97,684 < 98,97,684 < 98,97,686
2. 62,52,021 > 61,52,021 > 52,76,201 > 52,67,201 > 5,20,671
Exercise 1B
1.
a. 5,98,567 > 98,576
b. 4,56,98,786 > 97,89,999
c. 1,23,457 < 6,87,987
d. 13,78,654 > 13,76,689
e. 2,34,645 < 5,86,576
f. 9,03,412 < 9,13,689
2.
a. 7,80,643; 8,28,867; 80,25,957; 1,56,34,123
b. 2,23,927; 2,35,927; 3,72,227; 3,95,327
c. 25,45,068; 28,68,054; 39,54,068; 76,43,090
d. 8,53,60,295; 8,93,40,294; 9,45,04,949; 9,64,90,365

3. The greatest 7-digit number having four different digits is 99,99,876

4. Largest 6-digit number = 9,99,999
Largest 5-digit number = 99,999
Therefore, the total number of 6-digit numbers = (9,99,999 – 99,999) = 9,00,00

5. 3-digit numbers that can be formed using the digits 4, 5, and 6 without repetition of digits are:
456, 465, 546, 564, 645, 654
Descending order: 654, 645, 564, 546, 465, 456

Try This! (Page 15)

1. 10 crore = 10,00,00,000
So, there are 9 digits in 10 crores.
2. 10 lakhs = 10,00,000
10,00,000 can be written in international system as 1,000,000
1,000,000 = 1 million
So, in International system of numeration millions place is equivalent to ten lakhs place.
3. Indian system of numeration – 36,58,942
International system of numeration – 3,658,942

Exercise 1C
1.
a. 648,295,825 - Six hundred forty-eight million two hundred ninety-five thousand eight hundred
twenty-five
b. 59,379,356 - Fifty-nine million three hundred seventy-nine thousand three hundred fifty-six
c. 4,087,392,132 - Four billion eighty-seven million three hundred ninety-two thousand one hundred
thirty-two
d. 400,050,008 - Four hundred million fifty thousand eight
e. 345,000 - Three hundred forty-five thousand
f. 6,483,476 - Six million four hundred eighty-three thousand four hundred seventy-six
g. 5,930,086 - Five million nine hundred thirty-thousand eighty-six
h. 50,829,600 - Fifty million eight hundred twenty-nine thousand six hundred

2.
a. 2,34,47,146 - Two crore thirty-four lakh forty-seven thousand one hundred forty-six
b. 88,00,886 - Eighty-eight lakh eight hundred eighty-six
c. 9,38,70,042 - Nine crore thirty-eight lakh seventy thousand forty-two
d. 5,63,54,238 - Five crore sixty-three lakh fifty-four thousand two hundred thirty-eight
e. 54,39,760 - Fifty-four lakh thirty-nine thousand seven hundred sixty
f. 4,50,23,444 - Four crore fifty lakh twenty-three thousand four hundred forty-four
g. 10,00,643 - Ten lakh six hundred forty-three
h. 2,03,33,040 - Two crore three lakh thirty-three thousand forty
3. 10 lakhs = 10,00,000 = 1,000,000 = one million
Ten lakhs make a million.

4. 1 billion = 1,000,000,000 = 1000 million

One thousand million make a billion.

Exercise 1D
1.
a. 8 m = 800 cm.
800 cm < 850 cm
Thus, 8 m < 850 cm
b. 4 cm = 40 mm
40 mm < 450 mm
Thus, 4 cm < 450 mm
c. 3 km = 3000000 mm
3000000 mm > 3000 mm
Thus, 3 km > 3000 mm
d. 5000 cm = 50 m
5000 cm > 500 cm
Thus, 5000 cm > 5 m
e. 78 kg = 78000 g
780 g < 78000 g
Thus, 780 g < 78 kg
f. 1 g = 1000 mg
85000 mg = 85 g
g. 4000 g = 4 kg
4 kg < 30 kg
Thus, 4000 g < 30 kg
h. 3.5 kg = 3500 g
3500 g > 350 g
Thus, 3.5 kg > 350 g
i. 5 𝑙𝑙 = 5000 m𝑙𝑙
5000 m𝑙𝑙 > 500 m𝑙𝑙
Thus, 5 𝑙𝑙 > 500 m𝑙𝑙
j. 1 k𝑙𝑙 = 1000 𝑙𝑙 = 1,000,000 m𝑙𝑙
Thus, 3 k𝑙𝑙 = 3,000,000 m𝑙𝑙
k. 1 𝑙𝑙 = 1000 m𝑙𝑙
5000 m𝑙𝑙 = 5 𝑙𝑙
5 𝑙𝑙 < 20 𝑙𝑙
Thus, 5000 m𝑙𝑙 < 20 𝑙𝑙
l. 1 k𝑙𝑙 = 1000 𝑙𝑙
2000 𝑙𝑙 = 2 k𝑙𝑙
Thus, 2000 𝑙𝑙 = 2 k𝑙𝑙

Try This! (Page 18)

1. 9876 rounded off to nearest 10 = 9880
9876 rounded off to nearest 100 = 9900
9876 rounded off to nearest 1000 = 10,000

2. 6543 rounded off to nearest 100 = 6500

987 rounded off to nearest 100 = 1000
6500 × 900 = 65,00,000

Exercise 1E
1.
Add Round off Estimate Correct
47 + 68 50 + 70 120 115
67 + 43 70 + 40 110 110
47 + 43 50 + 40 90 90
44 + 66 40 + 70 110 110
77 + 83 80 + 80 160 160
34 + 71 30 + 70 100 105
63 + 78 60 + 80 140 141
89 + 24 90 + 20 110 113
34 + 67 30 + 70 100 101
78 + 84 80 + 80 160 162
2.
Subtract Round off Estimate Correct
88 − 35 90 − 40 50 53
85 − 38 90 − 40 50 47
43 − 22 40 − 20 20 21
78 − 39 80 − 40 40 39
47 − 23 50 − 20 30 24
76 − 28 80 − 30 50 48
94 − 37 90 − 40 50 57
67 − 39 70 − 40 30 28
84 − 37 80 − 40 40 47
93 − 49 90 − 50 40 44
3.
Multiply Round off Estimate Correct

39 × 8 40 × 8 320 312
53 × 8 50 × 8 400 424
91 × 4 90 × 4 360 364
82 × 4 80 × 4 320 328
73 × 7 70 × 7 490 511
69 × 6 70 × 6 420 414
36 × 4 40 × 4 160 144
63 × 4 60 × 4 240 252
74 × 7 70 × 7 490 518
76 × 7 80 × 7 560 532

4.
a. 4000 + 5000 = 9000
b. 5000 + 7000 = 12000
c. 9000 + 8000 = 17000
d. 3000 + 7000 = 10000
e. 2000 – 1000 = 1000
f. 9000 – 2000 = 7000
g. 7000 – 4000 = 3000
h. 8000 – 2000 = 6000
i. 2000 + 7000 = 9000
j. 9000 – 8000 = 1000

5.
a. 50 × 70 = 3500
b. 90 × 90 = 8100
c. 80 × 40 = 3200
d. 30 × 50 = 1500
e. 300 × 50 = 15000
f. 800 × 70 = 56000
g. 500 × 30 = 15000
h. 300 × 300 = 90000
i. 500 × 400 = 200000
j. 700 × 700 = 490000

Mental Maths
1. 2-digit numbers above 50 with sum of digits 12 are 57, 66, 75, 84, and 93.
2-digit numbers above 50 with difference of digits 4 are 51, 59, 62, 73, 84, and 95.
84 is the only number that exists in both the lists.
Therefore, the required number is 84.
2. 2-digit number below 50 with sum 13 is 49.
The product of the digits is 4 × 9 = 36.
Therefore, the number is 49.

3. Highest value of any digit can be 9.

The highest value of carry forward can be 1.
So, the highest value of (a + p) with carry 1 = 18 + 1 = 19
Therefore, the highest numerical value of ‘k’ in the problem is 1.

4. We know that 2 + 2 = 4, and 2 × 2 = 4.

Therefore, a two-digit number having same sum and product of the digits is 22.

5. 2-digit number that can be formed using the digits 7, 8, 0 are 70, 77, 80, 88, 78, 87.
Therefore, a total of six numbers can be formed.

Exercise 1F
1.
a. 15 = 10 + 5 = XV b. 21 = 20 + 1 = XXI
c. 25 = 20 + 5 = XXV d. 29 = 20 + 9 = XXIX
e. 38 = 30 + 8 = XXXVIII f. 42 = 40 + 2 = XLII
g. 47 = 40 +7 = XLVII h. 51 = 50 + 1 = LI
i. 62 = 60 + 2 = LXII j. 74 = 70 + 4 = LXXIV
k. 83 = 80 + 3 = LXXXIII l. 91 = 90 + 1 = XCI
m. 95 = 90 + 5 = XCV n. 97 = 90 + 7 = XCVII
o. 99 = 90 + 9 = XCIX p. 123 = 100 + 20 + 3 = CXXIII
q. 395 = 300 + 90 + 5 = CCCXCV r. 265 = 200 + 60 + 5 = CCLXV
s. 900 = 1000 – 100 = CM t. 1100 = 1000 + 100 = MC

2.
a. XXIII = 10 + 10 + 3 = 23 b. XXXI = 10 + 10 + 10 + 1 = 31
c. XL = 50 −1 = 40 d. XLIV = (50 – 10) + 4 = 44
e. LIV = 50 + 4 = 54 f. LXIII = 50 + 10 + 3 = 63
g. LXXVI = 50 + 20 + 5 + 1 = 76 h. LXXXI = 50 + 30 + 1 = 81
i. LXXXV = 50 + 30 + 5 = 85 j. XC = 100 – 10 = 90
k. XCIV = (100 – 10) + 4 = 94 l. XCVIII = (100 – 10) + 8 = 98
m. CCX = 100 + 100 + 10 = 210 n. CD = 500 – 100 = 400
o. DCC = 500 + 100 + 100 = 700
3.
b. XVV is meaningless as V cannot be repeated.
d. VCX is meaningless as the V is never written to the left of a symbol with greater value.
e. ILD is meaningless as L is never written to the left of a symbol with greater value.
Exercise 1G
1.
a. 6 × (10 + 8) = 6 × 18 = 108
b. 15 + (44 – 21) = 15 + 23 = 38
c. 14 × (20 – 8) = 14 × 12 = 168
d. 58 + (105 + 220) = 383
e. 252 – (305 – 135) = 252 – 170 = 82
f. 100 × (20 + 120) = 100 × 140 = 14000

2. Number of spectators on Monday = 36,731 + 38, 953 = 75,684

Number of spectators on Tuesday = 39,246 + 36,842 = 76,088
Difference in the number of spectators = 76,088 – 75,684 = 404
On Tuesday number of spectators are 404 more.

3. Number of capsules produced in a day = 24,800

Number of capsules packed in one container = 800
Number of containers needed to pack 24,800 capsules = 24,800 ÷ 800 = 31 containers
Cost of one container = ₹ 400
Cost of 31 containers = ₹ 400 × 31 = ₹12,400

4. Cost of one book = ₹ 500

Total amount collected by selling the books = ₹7,35,000
Number of books sold = 7,35,000 ÷ 500 = 1470
30 copies given to VIPs.
So, the total no. of books given out = 30 + 1470 = 1500

5. Total number of boxes = 65

Total number of packages = 65 × 80
Total number of toy soldiers = 65×80×15 = 78,000

6. The capacity of a water tank in a housing colony = 64 kilolitres = 64,000 litre

Total number of family = 80
Water used by families in 4 days = 80 × 4 = 320
Average quantity of water used per day per household = 64,000 ÷ 320 = 200 litre

7. One cookie needs total number of ingredients = 8 g

Number of cookies made by 44.4 kg of ingredients is = 44.4 kg ÷ 8 g
= 44400 ÷ 8 = 5550 cookies
8. Total number of participants = 62
Each participant ran = 750 m
Total distance travelled by them = 750 m × 62 = 46,500 m = 46.5 km

9. The price of half meter fabric is = ₹ 220

Then the price of one meter fabric is = 220 × 2 = ₹ 440
The length of fabric for ₹ 2640 = 2640 ÷ 440 = 6 m
Length of fabric she brought home = 6 m – 50 cm = 600 cm – 50 cm = 650 cm = 5 m 50 m

Higher Order Thinking Skills

1. Option 1: ₹ 1000 for 7 days.
Option 2: 10 + 20 + 40 + 80 + 160 + 320 + 640 = 1270
Second choice as Rashid will get 1270.

2. On 30th day, half of the pond was filled with lotus flowers. Therefore, on 31st it will be full, since the
number doubles every day.
Similarly, it was quarter full on 29th day.

Chapter Check-Up
1. 1 crore = 1,00,00,000
So, there are 7 zeros in one crore.

2. 1699 = 1000 + 600 + 90 + 9 = MDCXCIX

3.
a. 2,00,000 + 80,000 + 4000 + 200 + 30 + 1
b. 50,00,000 + 2,00,000 + 10,000 + 1000 + 500 + 60 + 8
c. 6,00,00,000 + 4,00,000 + 10,000 + 8000 + 500 + 10 + 7
d. 8,00,00,000 + 90,00,000 + 1,00,000 + 80,000 + 1000 + 200 + 10 + 3

4. a. 9,78,342 b. 36,23,047 c. 7,01,30,026 d. 3,00,04,301

5.
a. Place value of 2 in 14,12,326 is 2×1000 = 2000
b. Place value of 1 in 61,13,617 is 1×10,000 = 10,000
c. Place value of 2 in 7,28,05,821 is 2×10 = 20
d. Place value of 4 in 4,81,00,219 is 4×1,00,00,000 = 4,00,00,000

6. a. 62,45,635 b. 9,58,61,089
7. a. 41,36,15,128 b. 61,84,153 c. 79,00,12,900
8. a. 814,213,416 b. 92,813,241 c. 5,008,120,035
9.
a. 3228 + 572 = 3200 + 600 = 3800
b. 8010 – 2507 = 8000 – 2500 = 5500
c. 32 × 58 = 30 × 60 = 1800
d. 108 × 47 = 110 × 50 = 5500

10. a. 46 = 40 + 6 = XLVI b. 90 = 100 – 10 = XC

c. 120 = 100 + 20 = CXX d. 150 = 100 + 50 = CL
e. 260 = 200 + 60 = CCLX f. 350 = 300 + 50 = CCCL
g. 480 = 400 + 80 = CDLXXX h. 950 = (1000 – 100) + 50 = CML

11. Two years ago, the population of a district = 22,41,289

In two years, population of district increased by 48,000
Present population = 48,000 + 22,41,289 = 22,89,289

12. Distance covered from Delhi to Karnataka = 1927 km

Distance covered from Karnataka to Tamil Nadu = 633 km
Total distance covered if he retraces his journey = (1927 + 633) × 2 = 2560 × 2 = 5120 km

13. Number of days in September is 30.

Number of diyas made in the month of September = 4080
So, in one day the total number of diyas made by the potter = 4080 ÷ 30 = 136
In one-week, the total number of diyas made by the potter = 136 × 7 = 952 diyas

14. Largest 5-digit number is 99, 999.

Largest 4-digit number is 9,999.
Required number = 99,999 – 9,999 = 90,000.
So, there are 90,000 5-digit numbers in all.

15. Largest five-digit number using the digits 0, 8, 9 without repetition of the digits 8 and 9 = 98,000
Smallest five-digit number using the digits 0, 8, 9 without repetition of the digits 8 and 9 = 80,009
Difference = 98,000 – 80,009 = 17,991

Case-Study Based Question

16.
a. Indian 1,30,79,696 33,72,72,114 17,60,607
International 13,079,696 337,272,114 1,760,607
b. One crore thirty-lakh seventy-nine thousand six hundred ninety-six.
Thirty-three crore seventy-two lakh seventy-two thousand one hundred fourteen.
Seventeen lakh sixty thousand six hundred seven.
b. Thirteen million seventy-nine thousand six hundred ninety-six.
Three hundred thirty-seven million two hundred seventy-two thousand one hundred fourteen
One million seven hundred sixty thousand six hundred seven
17. Amount of Juice sold in the month of July = 5 kl 680 l = 5000 l + 680 l = 5680 l
Amount of Juice sold in the month of August = 6 kl 370 l = 6000 l + 370 l = 6370 l
Total juice sold in two month = 5680 + 6370 = 12,050 l = 12 kl 50 l

18. Total number of towers in complex = 27

16 towers have 14 floors, and 11 towers have 22 floors
Each floor has 8 flats.
Total number of flats = (16 ×14 × 8) + (11 × 22 × 8) = 3728 flats

Cross-Curricular Connect
a. Arrangements of the planets in ascending order based on their distance from the Sun:
Mercury (58,000,000 km) < Venus (108,000,000 km) < Earth (150,000,000 km) < Mars (228,000,000
km) < Jupiter (778,000,000 km).
b. The distance of Mars from the Sun in the Indian system of numeration = 22,80,00,000 km
c. The distance of Venus from the Sun in expanded form = 100,000,000 + 8,000,000
d. The distance between Mars and Earth = 228,000,000 – 150,000,000 = 78,000,000 km
The distance between Earth and Venus = 150,000,000 – 108,000,000 = 42,000,000 km
78,000,000 km > 42,000,000 km. Therefore, the distance between Mars and Earth is greater than the
distance between Earth and Venus.

Being Indian
Cost of bandhni fabric = ₹ 105 per metre
Cost of leheriya fabric = ₹ 95 per metre
Money earned by selling 185 m of bandhni fabric in a week = 105 × 185
Money earned by selling 185 m of leheriya fabric in a week = 95 × 185
Total money earned = 105 × 185 + 95 × 185 = 185 (105 + 95) = 185 × 200 = ₹ 37,000